In a large skillet, melt butter over medium heat. Add mushrooms, onions, garlic, and salt. Cook about 10 minutes or until juices released from mushrooms have evaporated. Add cognac; cook until cognac has evaporated.

On a lightly floured surface, roll out puff pastry to a 12-inch square. Cut into four 6-inch squares. Divide cheese among puff pastry squares, placing cheese in center of each square. Top with leftover chicken; top with mushroom mixture.

In a small bowl, lightly whisk together egg and the water. Brush the edges of the puff pastry squares with the egg mixture. Pull the corners of each pastry square to the center; pinch the edges closed. Place seam sides down, on a baking sheet. Chill in refrigerator for 15 minutes.

Preheat oven to 400 degrees. Remove Wellingtons from refrigerator and brush tops of pastries with the remaining egg mixture. Bake in preheated oven about 25 minutes or until puffed and golden brown.